# Conditions for additivity of the trace of projections

I came up with the following problem:

Let $\Pi_A$ and $\Pi_B$ be two projection operators on two disjoint subspaces of a certain Hilbert space $\mathcal H$ and let $\rho$ be unit trace, positive, hermitian matrix operating on $\mathcal H$.

Show that $Tr((\Pi_A+\Pi_B)\rho) = Tr((\Pi_A)\rho) + Tr((\Pi_B)\rho)$ whenever $\rho$ can be diagonalized in the same basis as $\Pi_A$ and $\Pi_B$.

Then, as far as I understand it follows by the linearity of the trace that the equality always holds even if $\rho$ can not be diagonalized in the same basis as the projection operators. Is that correct?

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Yes, if $\rho$ is trace-class, then $(\Pi_A+\Pi_B)\rho$, $\Pi_A\,\rho$, and $\Pi_B\,\rho$ are trace-class, and then the equality follows from the linearity of the trace.
You don't even need the operators to be projections: the assertion still holds for any bounded operators $A,B$.